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17
An InteriorPoint Method for Semidefinite Programming
, 2005
"... We propose a new interior point based method to minimize a linear function of a matrix variable subject to linear equality and inequality constraints over the set of positive semidefinite matrices. We show that the approach is very efficient for graph bisection problems, such as maxcut. Other appli ..."
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Cited by 202 (18 self)
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We propose a new interior point based method to minimize a linear function of a matrix variable subject to linear equality and inequality constraints over the set of positive semidefinite matrices. We show that the approach is very efficient for graph bisection problems, such as maxcut. Other applications include maxmin eigenvalue problems and relaxations for the stable set problem.
Semidefinite programs and combinatorial optimization (Lecture notes)
, 1995
"... this paper, we are only concerned about the last question, which can be answered using semidefinite programming. For a survey of other aspects of such geometric representations, see [64]. ..."
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Cited by 29 (1 self)
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this paper, we are only concerned about the last question, which can be answered using semidefinite programming. For a survey of other aspects of such geometric representations, see [64].
ConeLP's and Semidefinite Programs: Geometry and a Simplextype Method
, 1996
"... . We consider optimization problems expressed as a linear program with a cone constraint. ConeLP's subsume ordinary linear programs, and semidefinite programs. We study the notions of basic solutions, nondegeneracy, and feasible directions, and propose a generalization of the simplex method fo ..."
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Cited by 19 (2 self)
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. We consider optimization problems expressed as a linear program with a cone constraint. ConeLP's subsume ordinary linear programs, and semidefinite programs. We study the notions of basic solutions, nondegeneracy, and feasible directions, and propose a generalization of the simplex method for a large class including LP's and SDP's. One key feature of our approach is considering feasible directions as a sum of two directions. In LP, these correspond to variables leaving and entering the basis, respectively. The resulting algorithm for SDP inherits several important properties of the LPsimplex method. In particular, the linesearch can be done in the current face of the cone, similarly to LP, where the linesearch must determine only the variable leaving the basis. 1 Introduction Consider the optimization problem Min cx s:t: x 2 K Ax = b (P ) where K is a closed cone in R k , A 2 R m\Thetak ; b 2 R m ; c 2 R k : (P ) is called a linear program over a cone, or a coneLP. It m...
SEMIDEFINITE PROGRAMMING RELAXATIONS OF NONCONVEX PROBLEMS IN CONTROL AND COMBINATORIAL OPTIMIZATION
"... We point out some connections between applications of semidefinite programming in control and in combinatorial optimization. In both fields semidefinite programs arise as convex relaxations of NPhard quadratic optimization problems. We also show that these relaxations are readily extended to optimi ..."
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Cited by 17 (2 self)
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We point out some connections between applications of semidefinite programming in control and in combinatorial optimization. In both fields semidefinite programs arise as convex relaxations of NPhard quadratic optimization problems. We also show that these relaxations are readily extended to optimization problems over bilinear matrix inequalities.
Semidefinite Programming for Assignment and Partitioning Problems
, 1996
"... Semidefinite programming, SDP, is an extension of linear programming, LP, where the nonnegativity constraints are replaced by positive semidefiniteness constraints on matrix variables. SDP has proven successful in obtaining tight relaxations for NP hard combinatorial optimization problems of simpl ..."
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Cited by 13 (2 self)
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Semidefinite programming, SDP, is an extension of linear programming, LP, where the nonnegativity constraints are replaced by positive semidefiniteness constraints on matrix variables. SDP has proven successful in obtaining tight relaxations for NP hard combinatorial optimization problems of simple structure such as the maxcut and graph bisection problems. In this work, we try to solve more complicated combinatorial problems such as the quadratic assignment, general graph partitioning and set partitioning problems. A tight SDP relaxation can be obtained by exploiting the geometrical structure of the convex hull of the feasible points of the original combinatorial problem. The analysis of the structure enables us to find the socalled "minimal face" and "gangster operator" of the SDP. This plays a significant role in simplifying the problem and enables us to derive a unified SDP relaxation for the three different problems. We develop an efficient "partial infeasible" primaldual inter...
MaxMin Eigenvalue Problems, PrimalDual Interior Point Algorithms, and Trust Region Subproblems
 Optimization Methods and Software
, 1993
"... Two Primaldual interior point algorithms are presented for the problem of maximizing the smallest eigenvalue of a symmetric matrix over diagonal perturbations. These algorithms prove to be simple, robust, and efficient. Both algorithms are based on transforming the Technische Universitat Graz, I ..."
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Cited by 10 (4 self)
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Two Primaldual interior point algorithms are presented for the problem of maximizing the smallest eigenvalue of a symmetric matrix over diagonal perturbations. These algorithms prove to be simple, robust, and efficient. Both algorithms are based on transforming the Technische Universitat Graz, Institut fur Mathematik, Kopernikusgasse 24, A8010 Graz, Austria. Research support by Christian Doppler Laboratorium fur Diskrete Optimierung. y Program in Statistics & Operations Research, Princeton University, Princeton, NJ 08544. Research support by AFOSR through grant AFOSR910359. z Research support by the National Science and Engineering Research Council Canada. problem to one over the cone of positive semidefinite matrices. One of the algorithms does this transformation through an intermediate transformation to a trust region subproblem. This allows the removal of a dense row. Key words: Maxmin eigenvalue problems, trust region subproblems, Loewner partial order, primaldual ...
Explicit solutions for interval semidefinite linear programs
 Linear Algebra Appl
, 1996
"... We consider the special class of semidefinite linear programs (IV P) maximize traceCX subject to L ≼ A(X) ≼ U, where C, X, L, U are symmetric matrices, A is a (onto) linear operator, and ≼ denotes the Löwner (positive semidefinite) partial order. We present explicit representations for the general ..."
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Cited by 9 (2 self)
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We consider the special class of semidefinite linear programs (IV P) maximize traceCX subject to L ≼ A(X) ≼ U, where C, X, L, U are symmetric matrices, A is a (onto) linear operator, and ≼ denotes the Löwner (positive semidefinite) partial order. We present explicit representations for the general primal and dual optimal solutions. This extends the results for standard linear programming that appeared in BenIsrael and Charnes, 1968. This work is further motivated by the explicit solutions for a different class of semidefinite problems presented recently in Yang and Vanderbei, 1993.
Trust Regions and Relaxations for the Quadratic Assignment Problem
 IN QUADRATIC ASSIGNMENT AND RELATED PROBLEMS (NEW
, 1993
"... General quadratic matrix minimization problems, with orthogonal constraints, arise in continuous relaxations for the (discrete) quadratic assignment problem (QAP). Currently, bounds for QAP are obtained by treating the quadratic and linear parts of the objective function, of the relaxations, separat ..."
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Cited by 6 (5 self)
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General quadratic matrix minimization problems, with orthogonal constraints, arise in continuous relaxations for the (discrete) quadratic assignment problem (QAP). Currently, bounds for QAP are obtained by treating the quadratic and linear parts of the objective function, of the relaxations, separately. This paper handles general objectives as one function. The objectives can be both nonhomogeneous and nonconvex. The constraints are orthogonal or Loewner partial order (positive semidefinite) constraints. Comparisons are made to standard trust region subproblems. Numerical results are obtained using a parametric eigenvalue technique.